Some remarks on the hyperelliptic moduli of genus 3

TL;DR

This paper corrects previous inaccuracies in the equations of invariants for genus 3 hyperelliptic curves, providing explicit moduli equations and criteria for isomorphism over algebraically closed fields.

Contribution

It computes the correct invariant relations for binary octavics and derives explicit equations for the hyperelliptic moduli space of genus 3 curves.

Findings

Corrected the syzygies among invariants of binary octavics.

Derived necessary and sufficient conditions for isomorphism of genus 3 hyperelliptic curves.

Explicitly computed the hyperelliptic moduli equations in terms of absolute invariants.

Abstract

In 1967, Shioda \cite{Shi1} determined the ring of invariants of binary octavics and their syzygies using the symbolic method. We discover that the syzygies determined in \cite{Shi1} are incorrect. In this paper, we compute the correct equations among the invariants of the binary octavics and give necessary and sufficient conditions for two genus 3 hyperelliptic curves to be isomorphic over an algebraically closed field $k$, $\ch k \neq 2, 3, 5, 7$. For the first time, an explicit equation of the hyperelliptic moduli for genus 3 is computed in terms of absolute invariants.